The **Chebyshev polynomials** are two sequences of polynomials related to the cosine and sine functions, notated as and . They can be defined in several equivalent ways, one of which starts with trigonometric functions:

The **Chebyshev polynomials of the first kind** are defined by:

Similarly, the **Chebyshev polynomials of the second kind** are defined by:

That these expressions define polynomials in may not be obvious at first sight but follows by rewriting and using de Moivre's formula or by using the angle sum formulas for and repeatedly. For example, the double angle formulas, which follow directly from the angle sum formulas, may be used to obtain and , which are respectively a polynomial in and a polynomial in multiplied by . Hence and .

An important and convenient property of the *T _{n}*(

and

The Chebyshev polynomials *T _{n}* are polynomials with the largest possible leading coefficient whose absolute value on the interval [−1, 1] is bounded by 1. They are also the "extremal" polynomials for many other properties.

In 1952, Cornelius Lanczos showed that the Chebyshev polynomials are important in approximation theory for the solution of linear systems;^{[2]} the roots of *T _{n}*(

These polynomials were named after Pafnuty Chebyshev.^{[3]} The letter T is used because of the alternative transliterations of the name *Chebyshev* as *Tchebycheff*, *Tchebyshev* (French) or *Tschebyschow* (German).

The **Chebyshev polynomials of the first kind** are obtained from the recurrence relation:

The recurrence also allows to represent them explicitly as the determinant of a tridiagonal matrix of size :

The ordinary generating function for T_{n} is:

There are several other generating functions for the Chebyshev polynomials; the exponential generating function is:

The generating function relevant for 2-dimensional potential theory and multipole expansion is:

The **Chebyshev polynomials of the second kind** are defined by the recurrence relation:

Notice that the two sets of recurrence relations are identical, except for vs. . The ordinary generating function for U

and the exponential generating function is:

As described in the introduction, the Chebyshev polynomials of the first kind can be defined as the unique polynomials satisfying:

or, in other words, as the unique polynomials satisfying:

for

The polynomials of the second kind satisfy:

or

which is structurally quite similar to the Dirichlet kernel

(The Dirichlet kernel, in fact, coincides with what is now known as the Chebyshev polynomial of the fourth kind.)

An equivalent way to state this is via exponentiation of a complex number: given a complex number *z* = *a* + *bi* with absolute value of one:

Chebyshev polynomials can be defined in this form when studying trigonometric polynomials.

That cos *nx* is an nth-degree polynomial in cos *x* can be seen by observing that cos *nx* is the real part of one side of de Moivre's formula:

The real part of the other side is a polynomial in cos

Chebyshev polynomials can also be characterized by the following theorem:^{[5]}

If is a family of monic polynomials with coefficients in a field of characteristic such that and for all and , then, up to a simple change of variables, either for all or for all .

The Chebyshev polynomials can also be defined as the solutions to the Pell equation:

in a ring

The Chebyshev polynomials of the first and second kinds correspond to a complementary pair of Lucas sequences *Ṽ _{n}*(

It follows that they also satisfy a pair of mutual recurrence equations:

The second of these may be rearranged using the recurrence definition for the Chebyshev polynomials of the second kind to give:

Using this formula iteratively gives the sum formula:

while replacing and using the derivative formula for gives the recurrence relationship for the derivative of :

This relationship is used in the Chebyshev spectral method of solving differential equations.

Turán's inequalities for the Chebyshev polynomials are:^{[8]}

The integral relations are^{[7]}^{: 187(47)(48) }^{[9]}

where integrals are considered as principal value.

Different approaches to defining Chebyshev polynomials lead to different explicit expressions. The trigonometric definition gives an explicit formula as follows:

From this trigonometric form, the recurrence definition can be recovered by computing directly that the bases cases hold:

and

and that the product-to-sum identity holds:

Using the complex number exponentiation definition of the Chebyshev polynomial, one can derive the following expression:

The two are equivalent because .

An explicit form of the Chebyshev polynomial in terms of monomials *x*^{k} follows from de Moivre's formula:

where Re denotes the real part of a complex number. Expanding the formula, one gets:

The real part of the expression is obtained from summands corresponding to even indices. Noting and , one gets the explicit formula:

which in turn means that:

This can be written as a

with inverse:

where the prime at the summation symbol indicates that the contribution of

A related expression for *T*_{n} as a sum of monomials with binomial coefficients and powers of two is

Similarly, *U*_{n} can be expressed in terms of hypergeometric functions:

That is, Chebyshev polynomials of even order have even symmetry and therefore contain only even powers of x. Chebyshev polynomials of odd order have odd symmetry and therefore contain only odd powers of x.

A Chebyshev polynomial of either kind with degree n has n different simple roots, called **Chebyshev roots**, in the interval [−1, 1]. The roots of the Chebyshev polynomial of the first kind are sometimes called Chebyshev nodes because they are used as *nodes* in polynomial interpolation. Using the trigonometric definition and the fact that:

one can show that the roots of T

Similarly, the roots of U

The extrema of T

One unique property of the Chebyshev polynomials of the first kind is that on the interval −1 ≤ *x* ≤ 1 all of the extrema have values that are either −1 or 1. Thus these polynomials have only two finite critical values, the defining property of Shabat polynomials. Both the first and second kinds of Chebyshev polynomial have extrema at the endpoints, given by:

The extrema of on the interval where are located at values of . They are , or where , , and , i.e., and are relatively prime numbers.

Specifically,^{[12]}^{[13]} when is even:

- if , or and is even. There are such values of .
- if and is odd. There are such values of .

When is odd:

- if , or and is even. There are such values of .
- if , or and is odd. There are such values of .

This result has been generalized to solutions of ,^{[13]} and to and for Chebyshev polynomials of the third and fourth kinds, respectively.^{[14]}

The derivatives of the polynomials can be less than straightforward. By differentiating the polynomials in their trigonometric forms, it can be shown that:

The last two formulas can be numerically troublesome due to the division by zero (0/0 indeterminate form, specifically) at *x* = 1 and *x* = −1. By L'Hôpital's rule:

More generally,

which is of great use in the numerical solution of eigenvalue problems.

Also, we have:

where the prime at the summation symbols means that the term contributed by

Concerning integration, the first derivative of the T_{n} implies that:

and the recurrence relation for the first kind polynomials involving derivatives establishes that for

The last formula can be further manipulated to express the integral of T_{n} as a function of Chebyshev polynomials of the first kind only:

Furthermore, we have:

The Chebyshev polynomials of the first kind satisfy the relation:

which is easily proved from the product-to-sum formula for the cosine:

For

The polynomials of the second kind satisfy the similar relation:

(with the definition

for

which establishes the evenness or oddness of the even or odd indexed Chebyshev polynomials of the second kind depending on whether m starts with 2 or 3.

The trigonometric definitions of *T*_{n} and *U*_{n} imply the composition or nesting properties:^{[15]}

For

Since *T*_{m}(*x*) is divisible by x if m is odd, it follows that *T*_{mn}(*x*) is divisible by *T*_{n}(*x*) if m is odd. Furthermore, *U*_{mn−1}(*x*) is divisible by *U*_{n−1}(*x*), and in the case that m is even, divisible by *T*_{n}(*x*)*U*_{n−1}(*x*).

Both T_{n} and U_{n} form a sequence of orthogonal polynomials. The polynomials of the first kind T_{n} are orthogonal with respect to the weight:

on the interval [−1, 1], i.e. we have:

This can be proven by letting *x* = cos *θ* and using the defining identity *T*_{n}(cos *θ*) = cos(*nθ*).

Similarly, the polynomials of the second kind U_{n} are orthogonal with respect to the weight:

on the interval [−1, 1], i.e. we have:

(The measure √1 − *x*^{2} d*x* is, to within a normalizing constant, the Wigner semicircle distribution.)

These orthogonality properties follow from the fact that the Chebyshev polynomials solve the Chebyshev differential equations:

which are Sturm–Liouville differential equations. It is a general feature of such differential equations that there is a distinguished orthonormal set of solutions. (Another way to define the Chebyshev polynomials is as the solutions to those equations.)

The T_{n} also satisfy a discrete orthogonality condition:

where N is any integer greater than max(

For the polynomials of the second kind and any integer *N* > *i* + *j* with the same Chebyshev nodes *x*_{k}, there are similar sums:

and without the weight function:

For any integer *N* > *i* + *j*, based on the N zeros of *U*_{N }(*x*):

one can get the sum:

and again without the weight function:

For any given *n* ≥ 1, among the polynomials of degree n with leading coefficient 1 (monic polynomials):

is the one of which the maximal absolute value on the interval [−1, 1] is minimal.

This maximal absolute value is:

and |

Let's assume that *w _{n}*(

Define

Because at extreme points of T_{n} we have

From the intermediate value theorem, *f _{n}*(

By the equioscillation theorem, among all the polynomials of degree ≤ *n*, the polynomial f minimizes ‖ *f* ‖_{∞} on [−1, 1] if and only if there are *n* + 2 points −1 ≤ *x*_{0} < *x*_{1} < ⋯ < *x*_{n + 1} ≤ 1 such that | *f*(*x _{i}*)| = ‖

Of course, the null polynomial on the interval [−1, 1] can be approximated by itself and minimizes the ∞-norm.

Above, however, | *f* | reaches its maximum only *n* + 1 times because we are searching for the best polynomial of degree *n* ≥ 1 (therefore the theorem evoked previously cannot be used).

The Chebyshev polynomials are a special case of the ultraspherical or Gegenbauer polynomials , which themselves are a special case of the Jacobi polynomials :

Chebyshev polynomials are also a special case of Dickson polynomials:

In particular, when , they are related by and .

The curves given by *y* = *T*_{n}(*x*), or equivalently, by the parametric equations *y* = *T*_{n}(cos *θ*) = cos *nθ*, *x* = cos *θ*, are a special case of Lissajous curves with frequency ratio equal to n.

Similar to the formula:

we have the analogous formula:

For *x* ≠ 0:

and:

which follows from the fact that this holds by definition for

The first few Chebyshev polynomials of the first kind are OEIS: A028297

The first few Chebyshev polynomials of the second kind are OEIS: A053117

In the appropriate Sobolev space, the set of Chebyshev polynomials form an orthonormal basis, so that a function in the same space can, on −1 ≤ *x* ≤ 1, be expressed via the expansion:^{[16]}

Furthermore, as mentioned previously, the Chebyshev polynomials form an orthogonal basis which (among other things) implies that the coefficients *a*_{n} can be determined easily through the application of an inner product. This sum is called a **Chebyshev series** or a **Chebyshev expansion**.

Since a Chebyshev series is related to a Fourier cosine series through a change of variables, all of the theorems, identities, etc. that apply to Fourier series have a Chebyshev counterpart.^{[16]} These attributes include:

- The Chebyshev polynomials form a complete orthogonal system.
- The Chebyshev series converges to
*f*(*x*) if the function is piecewise smooth and continuous. The smoothness requirement can be relaxed in most cases – as long as there are a finite number of discontinuities in*f*(*x*) and its derivatives. - At a discontinuity, the series will converge to the average of the right and left limits.

The abundance of the theorems and identities inherited from Fourier series make the Chebyshev polynomials important tools in numeric analysis; for example they are the most popular general purpose basis functions used in the spectral method,^{[16]} often in favor of trigonometric series due to generally faster convergence for continuous functions (Gibbs' phenomenon is still a problem).

Consider the Chebyshev expansion of log(1 + *x*). One can express:

One can find the coefficients *a _{n}* either through the application of an inner product or by the discrete orthogonality condition. For the inner product:

which gives:

Alternatively, when the inner product of the function being approximated cannot be evaluated, the discrete orthogonality condition gives an often useful result for *approximate* coefficients:

where δ

For any N, these approximate coefficients provide an exact approximation to the function at x

This allows us to compute the approximate coefficients a_{n} very efficiently through the discrete cosine transform:

To provide another example:

The partial sums of:

are very useful in the approximation of various functions and in the solution of differential equations (see spectral method). Two common methods for determining the coefficients a

As an interpolant, the N coefficients of the (*N* − 1)st partial sum are usually obtained on the Chebyshev–Gauss–Lobatto^{[17]} points (or Lobatto grid), which results in minimum error and avoids Runge's phenomenon associated with a uniform grid. This collection of points corresponds to the extrema of the highest order polynomial in the sum, plus the endpoints and is given by:

An arbitrary polynomial of degree N can be written in terms of the Chebyshev polynomials of the first kind.^{[9]} Such a polynomial *p*(*x*) is of the form:

Polynomials in Chebyshev form can be evaluated using the Clenshaw algorithm.